Brownian Motion Calculus

Brownian Motion Calculus......Page 3
Contents......Page 9
Preface......Page 15
1.1 Origins......Page 19
1.2 Brownian Motion Specification......Page 20
1.3 Use of Brownian Motion in Stock Price Dynamics......Page 22
1.4 Construction of Brownian Motion from a Symmetric Random Walk......Page 24
1.5 Covariance of Brownian Motion......Page 30
1.6 Correlated Brownian Motions......Page 32
1.7 Successive Brownian Motion Increments......Page 34
1.7.1 Numerical Illustration......Page 35
1.8.1 Simulation of Brownian Motion Paths......Page 37
1.8.2 Slope of Path......Page 38
1.8.3 Non-Differentiability of Brownian Motion Path......Page 39
1.8.4 Measuring Variability......Page 42
1.9 Exercises......Page 44
1.10 Summary......Page 47
2.1 Simple Example......Page 49
2.2 Filtration......Page 50
2.3 Conditional Expectation......Page 51
2.3.1 General Properties......Page 52
2.4.1 Martingale Construction by Conditioning......Page 54
2.6.1 Sum of Independent Trials......Page 55
2.6.2 Square of Sum of Independent Trials......Page 56
2.6.4 Random Process B(t)......Page 57
2.6.6 Frequently Used Expressions......Page 58
2.7 Process of Independent Increments......Page 59
2.9 Summary......Page 60
3.1 How a Stochastic Integral Arises......Page 63
3.2 Stochastic Integral for Non-Random Step-Functions......Page 65
3.3 Stochastic Integral for Non-Anticipating Random Step-Functions......Page 67
3.4 Extension to Non-Anticipating General Random Integrands......Page 70
3.5 Properties of an It ¯ o Stochastic Integral......Page 75
3.6 Significance of Integrand Position......Page 77
3.7 It ¯ o integral of Non-Random Integrand......Page 79
3.8 Area under a Brownian Motion Path......Page 80
3.9 Exercises......Page 82
3.10 Summary......Page 85
3.11 A Tribute to Kiyosi It ¯ o......Page 86
Acknowledgment......Page 90
4.1 Stochastic Differential Notation......Page 91
4.2 Taylor Expansion in Ordinary Calculus......Page 92
4.3 It ¯ o’s Formula as a Set of Rules......Page 93
4.4.1 Frequent Expressions for Functions of Two Processes......Page 96
4.4.2 Function of Brownian Motion f [B(t )]......Page 98
4.4.3 Function of Time and Brownian Motion f [t ,B(t )]......Page 100
4.4.4 Finding an Expression for T t =0 B(t ) dB(t )......Page 101
4.4.5 Change of Numeraire......Page 102
4.4.6 Deriving an Expectation via an ODE......Page 103
4.5 L´ evy Characterization of Brownian Motion......Page 105
4.6 Combinations of Brownian Motions......Page 107
4.7 Multiple Correlated Brownian Motions......Page 110
4.8 Area under a Brownian Motion Path – Revisited......Page 113
4.9 Justification of It ¯ o’s Formula......Page 114
4.10 Exercises......Page 118
4.11 Summary......Page 119
5.1 Structure of a Stochastic Differential Equation......Page 121
5.2 Arithmetic Brownian Motion SDE......Page 122
5.3 Geometric Brownian Motion SDE......Page 123
5.4 Ornstein–Uhlenbeck SDE......Page 126
5.5 Mean-Reversion SDE......Page 128
5.7 Expected Value of Square-Root Diffusion Process......Page 130
5.8 Coupled SDEs......Page 132
5.10 General Solution Methods for Linear SDEs......Page 133
5.11 Martingale Representation......Page 138
5.12 Exercises......Page 141
5.13 Summary......Page 142
6 Option Valuation......Page 145
6.1 Partial Differential Equation Method......Page 146
6.2 Martingale Method in One-Period Binomial Framework......Page 148
6.3 Martingale Method in Continuous-Time Framework......Page 153
6.4 Overview of Risk-Neutral Method......Page 156
6.5.1 Digital Call......Page 157
6.5.2 Asset-or-Nothing Call......Page 159
6.5.3 Standard European Call......Page 160
6.6.1 Feynman-Ka c Link between PDE Method and Martingale Method......Page 162
6.6.2 Multi-Period Binomial Link to Continuous......Page 164
6.7 Exercise......Page 165
6.8 Summary......Page 166
7.1 Change of Discrete Probability Mass......Page 169
7.2 Change of Normal Density......Page 171
7.3 Change of Brownian Motion......Page 172
7.4 Girsanov Transformation......Page 173
7.5 Use in Stock Price Dynamics – Revisited......Page 178
7.6 General Drift Change......Page 180
7.7 Use in Importance Sampling......Page 181
7.8 Use in Deriving Conditional Expectations......Page 185
7.9 Concept of Change of Probability......Page 190
7.10 Exercises......Page 192
7.11 Summary......Page 194
8.1.1 In Discrete Time......Page 197
8.1.2 In Continuous Time......Page 200
8.2.1 Dynamics of Forward Price of a Bond......Page 202
8.2.2 Dynamics of Forward Price of any Traded Asset......Page 203
8.3.1 Exchange Option......Page 205
8.3.3 European Call under Stochastic Interest Rate......Page 206
8.4 Relating Change of Numeraire to Change of Probability......Page 208
8.5 Change of Numeraire for Geometric Brownian Motion......Page 210
8.6 Change of Numeraire in LIBOR Market Model......Page 212
8.7 Application in Credit Risk Modelling......Page 216
8.8 Exercises......Page 218
8.9 Summary......Page 219
ANNEXES......Page 221
A.1 Moment Generating Function and Moments of Brownian Motion......Page 223
A.3 Brownian Motion Reflected at the Origin......Page 226
A.4 First Passage of a Barrier......Page 232
A.5 Alternative Brownian Motion Specification......Page 234
B.1 Riemann Integral......Page 239
B.2 Riemann–Stieltjes Integral......Page 244
B.3 Other Useful Properties......Page 249
B.4 References......Page 252
C.1 Quadratic Variation......Page 253
C.2 First Variation......Page 256
D.1 Distance between Points......Page 257
D.2 Norm of a Function......Page 260
D.4 Norm of a Random Process......Page 262
D.5 Reference......Page 264
E.1 Central Limit Theorem......Page 265
E.2 Mean-Square Convergence......Page 266
E.3 Almost Sure Convergence......Page 267
E.5 Summary......Page 268
Answers to Exercises......Page 271
References......Page 317
Index......Page 321